cubeful distribution
Unknown
I rolled out the starting position 50.000 times using Jellyfish
Level 5 to produce the table below to show cubeful distribution.
Cube Level 1 2 4 8 16 32 Total
Cashes 37.83 13.54 1.78 0.28 0.03 0.01 53.47
Single 26.63 5.03 0.59 0.06 32.31
Gammon 1.45 10.9 1.17 0.10 0.01 13.63
Backgammon 0.07 0.46 0.04 0.01 0.01 0.59
100.0
Just in case it's not clear. Cashes are games that end with a cube
turn.
Singles Gammons and Backgammons are games played to conclusion.
I'm not technically minded enough to make use of these numbers, but
would they maybe affect match equity tables etc or could they shed
any sort of light on recube vig?
Roland Sutter
Stig Eide
Just a question: How can there be 27% single games with the cube at 1?
Shouldn't this only happen in some very rare games?
Stig Eide
In article <376c496f...@news.which.net>,
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Unknown
wrote:
>Hi. Thanks for the numbers.
>Just a question: How can there be 27% single games with the cube at 1?
>Shouldn't this only happen in some very rare games?
>Stig Eide
Oh Sh**
Sorry I aligned the table wrong here is the corrected version!
Cube Level 1 2 4 8 16 32 Total
Cashes 37.83 13.54 1.78 0.28 0.03 0.01 53.47
Single 0.00 26.63 5.03 0.59 0.06 32.31
Gammon 1.45 10.90 1.17 0.10 0.01 13.63
Backgammon 0.07 0.46 0.04 0.01 0.01 0.59
100.00
Roland Sutter
Gary Wong
> I rolled out the starting position 50.000 times using Jellyfish
> Level 5 to produce the table below to show cubeful distribution.
>
> Cube 1 2 4 8 16 32 Total
>
> Cashes 37.83 13.54 1.78 0.28 0.03 0.01 53.47
> Single 0.00 26.63 5.03 0.59 0.06 32.31
> Gammon 1.45 10.90 1.17 0.10 0.01 13.63
> Backgammon 0.07 0.46 0.04 0.01 0.01 0.59
> 39.35 51.53 8.02 0.98 0.11 0.01 100.00
Thanks very much for that data!
> I'm not technically minded enough to make use of these numbers, but
> would they maybe affect match equity tables etc or could they shed
> any sort of light on recube vig?
Unfortunately the results are probably inapplicable to match play (where
cube behaviour is likely to vary drastically from money play depending
on the score) but they ought to help us model certain features of money
games.
I have attempted to produce a simple Markov process which would produce
a distribution similar to that above. It is:
If the cube is centred:
- it is offered and dropped with probability 0.378 (terminal)
- it is offered and taken with probability 0.606 (go to the state below)
- a player wins a gammon with probability 0.015 (terminal)
- a player wins a backgammon with probability 0.07 (terminal)
If the cube is owned:
- it is offered and dropped with probability 0.220 (terminal)
- it is offered and taken with probability 0.152 (remain in this state)
- a player wins a single game with probability 0.455 (terminal)
(These wins can reasonably expected to belong to the player who last
turned the cube; we presume the cube holder can always reach a point
where she has a correct double before she wins a single game).
- a player wins a gammon with probability 0.172 (terminal)
- a player wins a backgammon with probability 0.001 (terminal)
(It's hard to determine who the gammon and backgammon wins belong to.
Certainly the vast majority will be the player who turned the cube, but
there will be occasions where the cube owner becomes too good to double
and goes on to win a gammon or backgammon without having had a correct
double.)
If two players happen to play with cube behaviour described by the above
model, then we can make the following observations about the results of
their games:
- Most (62%) initial doubles are takes. This value agrees reasonably
closely to an earlier result at:
http://x24.deja.com/=dnc/getdoc.xp?AN=376082872
- Most (59%) redoubles are drops.
- The final cube value follows the following (geometric-like) distribution:
p(X=1) = 0.394
p(X=2^n) = 0.514 x 0.152^{n-1} for n = 1, 2, 3...
- The cube is turned and accepted 0.71 times per game on average; the
average final value of the cube is 1.87.
- A single game is worth 2.2 points on average; the standard deviation is
3.0. I have previously assumed 2 and 3 respectively for these parameters
in my own games.
Those statistics do not change a significant amount if the Jacoby rule
is used in interpreting the score (naturally, the parameters of the model
might change depending on the use of the rule).
- Assuming all gammon and backgammon wins belong to the player who last
cubed, a double/take leaves the taker with an average normalised cubeful
equity of -0.446. (If all doubles were perfectly efficient, this figure
would be -0.5.)
With a few more assumptions it may be possible to come up with other
statistics on cube efficiencies, recube vig, etc. I don't have time
to think about it right now, though -- more another day perhaps.
Cheers,
Gary.
--
Gary Wong, Department of Computer Science, University of Arizona
ga...@cs.arizona.edu http://www.cs.arizona.edu/~gary/